# Full text of "Treatise On Analysis Vol-Ii"

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```5    UPPER AND LOWER INTEGRALS       115

defined, and is denoted by £/„; also it belongs to £ (12.7.6). If we apply

(13.5.2) to the sequence of partial sums  ^fn9 and (13.5.3) to each of the

»= i
terms in this partial sum, we obtain the following corollary:

(13.5.4)    If(fn)nzi w & sequence of functions ^0 belonging to «/, then

(00         \            00

!/„ = lA/.).
„=*!      /      „=!

Now consider an arbitrary mapping / of X into R. There always exist
functions heJ^ such that /z^/, for example the constant function equal
to +00. Put

(13.5.5)                                   M*(/) =     inf   )u*(fc);

this number £**(/) is called the upper integral of / with respect to the
measure ju.

If /e «/ it is clear that this definition agrees with the preceding one. Here
the value of /**(/) can be any element of R. The relation /^ g implies
/**(/) ^ M*(#)- F°r anY scalar a > 0 we have ti*(af) = <3ju*(/).

(1 3.5.6)   If the sumft +/2 o/ /wo mappings ofX into R /51 defined at all points
o/X, and if n*^) > — oo and /J,*(f2) > — oo,

This is obvious if one of the numbers n*(fi)> /**(/2) 1S +°°* If not,
given any a > /^*(/!) and b > ju*(/2), there exist hl9 h2 in */ such that
A ^ /zi, /2 ^ ^2 and ^(/zj) ^ a, ju*(/z2) ^ *• It follows that A4 + A2 ^/i +/2
and /X*(/ZA + /z2) ^ a + £ by (13.5.3). Hence the result.

(13.5.7)   If(fn) is any increasing sequence of mappings o/X into E such that
> — oo for all sufficiently large n, then

M*(sup/n\

\   n        /

= sup /**(/„) = lim

The inequality ju*/sup/w\ ^ sup u*(fn) is clear. Let us prove the reverse
inequality.  We may assume  that  sup ju*(/M) < +00,  otherwise there is

n

nothing to prove. By hypothesis, we may therefore assume that sup /x*(/n)

n

and all the ju*(/n) art finite. For each e > 0, we shall show that there exists an>  the
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